Answer
$0.1732$
Work Step by Step
We have:
$\nabla h=(-\pi y \sin (\pi xy) +z^2) i-(-\pi x \sin (\pi xy) )j+2xzk=(-\pi (-1) \sin (\pi (1)) +1^2) i+\pi (-1\sin \pi )j+2(1)k =\pi \sin \pi i+\pi \sin \pi j =i+2k$
$\implies \nabla h(-1,-1,-1)=i+2k $
Here, $v=i+j+k$
and $u=\dfrac{v}{|v|}=\dfrac{i+j+k}{\sqrt{(1)^2+(1)^2+(1)^2}}=\dfrac{1}{\sqrt 3} (i+j+k)$
and $\nabla h \cdot u=(i+2k) \cdot \dfrac{1}{\sqrt 3} (i+j+k)=\sqrt 3$
Now, $dh=ds(\nabla h \cdot u) \implies (0.1) (\dfrac{3}{\sqrt 3})$
or, $dh=0.1732$
